To think that Hayek favored socialized medicine is absurd. Hayek was not an anarchist. Government was necessary in society. Hayek recognized that and so it does have role to play. Hayek thought that US Government has over stepped it authority regarding healthcare.

F.A. Hayek stated in the preface to the 1976 edition of The Road to Serfdom. The problems of the American healthcare system are caused entirely by the fact that the government subjects the system to massive interventions, some of which are fascist in nature, while others are socialist.